Optimal. Leaf size=68 \[ \frac{2 b^2 \sin (c+d x) \sqrt{b \sec (c+d x)}}{d}-\frac{2 b^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}} \]
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Rubi [A] time = 0.0445958, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {16, 3768, 3771, 2639} \[ \frac{2 b^2 \sin (c+d x) \sqrt{b \sec (c+d x)}}{d}-\frac{2 b^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \cos (c+d x) (b \sec (c+d x))^{5/2} \, dx &=b \int (b \sec (c+d x))^{3/2} \, dx\\ &=\frac{2 b^2 \sqrt{b \sec (c+d x)} \sin (c+d x)}{d}-b^3 \int \frac{1}{\sqrt{b \sec (c+d x)}} \, dx\\ &=\frac{2 b^2 \sqrt{b \sec (c+d x)} \sin (c+d x)}{d}-\frac{b^3 \int \sqrt{\cos (c+d x)} \, dx}{\sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}\\ &=-\frac{2 b^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 b^2 \sqrt{b \sec (c+d x)} \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0312653, size = 50, normalized size = 0.74 \[ \frac{2 b^2 \sqrt{b \sec (c+d x)} \left (\sin (c+d x)-\sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.218, size = 324, normalized size = 4.8 \begin{align*} 2\,{\frac{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{5}} \left ( i\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) -i\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) \cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +i\sin \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) -i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sin \left ( dx+c \right ) -\cos \left ( dx+c \right ) +1 \right ) \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{5/2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (d x + c\right )\right )^{\frac{5}{2}} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \sec \left (d x + c\right )} b^{2} \cos \left (d x + c\right ) \sec \left (d x + c\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (d x + c\right )\right )^{\frac{5}{2}} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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